I don't even think this is technically possible, considering that Pokemon species are hard-coded into the game, but for the purposes of illustrating why this would be a bad idea I'll assume by miraculous chance that the game can manage it.
First of all, this is a feature from the TCG, which has a radically different system than that found in the main series of games. First of all (I'm hazy on the details but I have the core concepts), each Pokemon can only sport a single type, and there are some seven to nine types in the game, with resistances and weaknesses defined for each Pokemon. Consider, on the other hand, that in the main series of the game that weaknesses are defined for each type and stack upon each other, but even more pressing is the much greater variability in typings of Pokemon.
From seven to nine possible typings in the TCG, you can raise that number to 289 type combinations and you'll get the idea that determining which one to use won't be the easiest of feats. Moreover, many of those combos would worsen off the Pokemon that possessed them. Say, for instance, you happen to get a Water/Poison Slugma: so what? How do you expect a Pokemon that gets no good Water or Poison moves to be able to use its STAB to its full advantage? Simply put, Pokemon like this would be the Sneasels of the game, having intriguing typings that they have no potential to use.
In all, this would be a programmer's nightmare that is not worth the effort of implementation and will likely not be given any appreciation from the fans.